Research On Melnikov Method Of Chaos In Discontinuous Random Systems

Chaotic dynamics is an important branch of complexity science and a popular subject in the past thirty years,and has shown strong vitality in many fields.The random Melnikov method,as the most commonly used analytical method for studying chaotic motion of stochastic systems,has attracted more and more scholars' attention.The main purpose of this paper is to develop the traditional random Melnikov method suitable for smooth random systems,making it suitable for discontinuous systems under random excitation.First,the basic idea of the random Melnikov method is used,that is,the random Melnikov process function is obtained by measuring the distance between the stable and unstable manifolds of a perturbed random system.Then,by establishing the mean square criterion of chaos occurrence in the statistical sense,a threshold function for chaos occurrence in random systems is established.Finally,the results are obtained theoretically and numerically.In Chapter 2,it is first assumed that there is a monotonic function that divides the entire plane into two parts,and each region is described by a smooth system.Such a system is a piecewise continuous system,and there is a homoclinic orbit across the tangent plane.When excited by bounded noise,the system will bifurcate at a certain moment.At this time,the corresponding random Melnikov function is obtained by measuring the distance between the stable and unstable manifolds.When the random Melnikov function has simple zeros,chaotic phenomena may occur in random systems.According to this principle,the statistical mean square criterion for random systems is obtained,and the necessary conditions for predicting chaos in random systems are given from the perspective of energy.In Chapter 3,the random Melnikov chaos of a typical symmetric discontinuous random system with two tangent planes is studied.Based on the basic idea of the Melnikov method,a random Melnikov function is obtained,which can be divided into certain terms and random terms.According to Chapter 2,the specific expression of the response function can be obtained,and the corresponding variance can be obtained after transformation,so the mean square criterion formula is obtained.It can be seen from this formula that the chaos threshold is a function of noise intensity and damping,and that changes in noise intensity and damping have a certain effect on the threshold.Later,Poincarémapping and 0-1 tests are used to verify the previous correlation results.Through the research,it is found that to a certain extent,the noise intensity can not only generate or strengthen chaos,but also suppress chaos,which is consistent with the theoretical results.